To understand why sine and cosine do not have asymptotes but the others do, we need to remember our Unit Circle ratios.
Since, r is a constant that will always be 1, sine and cosine will always be real numbers. However, that is not the case with the other trig functions. There are times when sine (y-value) or cosine (x-value) will equal 0. That would make their ratios have a denominator of 0, causing them to become undefined because it's not impossible to have something over nothing. As a result, for tangent, cotangent, cosecant, and secant, will have asymptotes, lines that the graph can never cross. Where these asymptotes are will be dictated by their denominator in the ratio. For example, tangent and secant both have a denominator of x (cosine) so wherever cosine is equal 0, there will be an asymptote. For cosecant and cotangent, they have a denominator of y, so their asymptotes will be wherever y (sine) is equal to 0. That is why the 4 graphs will always look like they're approaching the asymptotes, but they can never touch or cross it because that is where they are undefined. Sine and cosine are under no such restrictions, since their denominator will always be 1, making it impossible for their ratios to become undefined at any value.